Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
680 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#99 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C25×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#1351 | C2 | semidirect |
| 5. C23×Q8×C53 | C4 | semidirect |
| 11. C53×#393 | C4 | semidirect |
| 15. C2×#95×C53 | C4 | semidirect |
| 150. #11808 | C53 | direct (N × H) |
| 186. #1351 | C2×C53 | semidirect |
| 196. C23×Q8 | C4×C53 | semidirect |
| 202. #393 | C4×C53 | semidirect |
| 206. C2×#95 | C4×C53 | semidirect |
| 211. C53 | #11808 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#393 | 2 |
| C53×#1819 | 2 |
| C53×#1462 | 2 |
| C53×#1351 | 2 |
| #11808 | 53 |